## A theorem of validity for Edgeworth expansions.(English)Zbl 0595.62119

A method for approximating the exact densities of the estimators by Edgeworth expansion is derived and applied to an autoregressive equation which frequently arises in economic models. Let c(p,T) be a vector of statistics where p denotes the sample moments and T the sample size. We need to derive the exact distribution function of $g(x)=\sqrt{T}(c(p,T)-c(\mu,T)),$ where $$\mu$$ is the expectation of p. Since it is impossible in general to derive the exact distribution, the authors propose its Edgeworth approximation, which is proved under certain stated regularity and continuity conditions to have an error of order $$o(T^{-r/2})$$, where the Edgeworth approximation is of order r. This is applied to a linear dynamic equation $y_ t=\sum^{s}_{j=1}a_ jy_{t-j}+\sum^{K}_{j=1}b_ jx_{tj}+u_ t$ where $$y_ t$$ is an endogenous variable, $$y_{t-j}$$ are lagged values of $$y_ t$$, $$x_{tj}$$ are values of K exogenous variables and $$u_ t$$ is an error process with mean zero and variance $$\sigma^ 2$$.
Reviewer: J.K.Sengupta

### MSC:

 62P20 Applications of statistics to economics 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distribution of statistics
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